Simple games cover voting systems in which a single alternative, such as a bill or an
amendment, is pitted against the status quo. A simple game or a yes-no voting system is a
set of rules that specifies exactly which collections of “yea” votes yield passage of the
issue at hand. Each of these collections of “yea” voters forms a winning coalition. We are
interested in performing a complexity analysis on problems defined on such families of
games. This analysis as usual depends on the game representation used as input. We
consider four natural explicit representations: winning, losing, minimal winning, and
maximal losing. We first analyze the complexity of testing whether a game is simple and
testing whether a game is weighted. We show that, for the four types of representations,
both problems can be solved in polynomial time. Finally, we provide results on the
complexity of testing whether a simple game or a weighted game is of a special type. We
analyze strongness, properness, weightedness, homogeneousness, decisiveness and
majorityness, which are desirable properties to be fulfilled for a simple game. Finally,
we consider the possibility of representing a game in a more succinct and natural way and
show that the corresponding recognition problem is hard.

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